All Questions
13 questions
1
vote
0
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66
views
CLT of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional random matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\...
- 284
1
vote
1
answer
84
views
Asymptotic property of the left singular vectors of i.i.d. data matrix
Let $\mathbf{X}$ be $(n \times p)$-dimensional data matrix ($n > p$) whose rows $\mathbf{x}_i$ are i.i.d. with some finite moments:
$$
\mathbf{X}^\top = [\mathbf{x}_1, \ldots \mathbf{x}_n]^\top.
...
- 284
0
votes
1
answer
103
views
Approximating the expectation of trace inverse of random Gaussian combination
Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. Has anyone studied this expectation in asymptotics $$E_{A}(\mathrm{Tr}( (A^T ...
- 25
2
votes
1
answer
231
views
Trace inverse of random PSD matrix?
Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. I am interested in the expectation of $$E_{A}(\mathrm{Tr}( (A^T A + \lambda \...
- 25
1
vote
1
answer
160
views
Estimates of product of eigenvalues gaps for Wigner matrices
Let $W_n$ be an $n\times n$ Wigner matrix$^{1}$, and let $\lambda_1\le \lambda_2\le \cdots \le \lambda_n$ be the eigenvalues of $\frac{W_n}{\sqrt{n}}$.
My question. For any fixed $i\in\{1,\dots,n\}$, ...
- 2,712
2
votes
1
answer
1k
views
Marchenko-Pastur Law under general covariance structure
Let $x_1,...,x_n\in\mathbb{R}^p$ be i.i.d. random vectors with mean 0 and covariance $\Sigma_p$. Let $S_{n,p}=\sum_{i=1}^nx_ix_i^T/n$ be the sample covariance. We consider the asymptotics of the ...
- 1,495
4
votes
1
answer
276
views
Berry–Esseen bound for operator norm of matrix averages
Is there a Berry–Esseen bound for operator norm of an average of independent random matrices?
Suppose $A_1, \dotsc, A_n$ are independent matrices with $\mathbb{E}[A_i] = I$ (the identity matrix). Is ...
- 51
7
votes
1
answer
857
views
Trace of inverse of random positive-definite matrix in high dimension?
Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
- 2,306
3
votes
1
answer
178
views
Tail probability of random projection
Suppose $v\in R^n$ is a constant unit vector. $P_l$ is a random projection matrix to an $l$ dimensional subspace of $R^n$ which is uniformly sampled from $G(l,R^n)$ which is the collection of all $l$-...
- 1,495
6
votes
2
answers
203
views
Asymptotic behavior of the ratio between the largest two singular values of product of i.i.d. random complex matrices
Let $A_n$ be the matrix product of $n$ i.i.d. N-by-N random complex matrices. The matrix distribution is not fixed and can be tuned to suit specific solution if needed, as long as it's not too "...
- 63
4
votes
1
answer
207
views
Upper bound on the number of binary matrices with small rank
I'm looking for the tightest upper bound on the number of different binary matrices $A \in {\{-1,1\}^{m \times n}}$ for which $\mathrm{rank}(A)\leq r$. I'm interested in the regime $1 \ll r \ll m \...
- 968
3
votes
0
answers
396
views
Sum of random permutation matrices
Let $A$ be a uniformly random $k\times k$ permutation matrix, and $A_1,\ldots, A_m$ be the $m$ independent copies of $A$. Here the uniform distribution is with respect to the $k!$ possible permutation ...
- 1,127
7
votes
0
answers
396
views
Stable distributions for Lindeberg exchange strategy?
Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random ...
- 2,350